Finite disjoint-union conjecture for finitely presented monoids of linear growth

Let $S$ be a finitely presented monoid of linear growth. A free sandwich is a subset of the form $p\langle q\rangle r$, where $p,q,r$ are words and $\langle q\rangle$ denotes the submonoid generated by $q$. Finite disjoint-union conjecture. The monoid $S$ is a finite disjoint union of free sandwiches and a finite set. The result is presented as a weaker version of Ufnarovski's automaton conjecture. The paper notes that the analogous assertion is known for finitely generated semigroups, while for finitely presented monoids the question remains open; examples with infinitely many defining relations show that disjointness can fail without finite presentability.